Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Coating — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Twice as fast is not twice as thick. The film a plate carries out of water, against the speed it is withdrawn at, both logarithmic. The slope is exactly two-thirds, so doubling the speed thickens the film by 58.7 per cent and never by more. The open marks are outside the range the derivation holds in — above a capillary number of about a hundredth the film is no longer thin against the capillary length, and the measured thickness leaves this line.

    What a plate takes with it

    Pull a plate out of a bath and it comes out wet. How wet is not set by the plate, the bath or how much liquid there is, but by a competition in a region a fraction of a millimetre long that nobody looking at the plate can see — and the film goes as the two-thirds power of the speed, never as the first.

    part 1 · viscous
  2. Below the capillary length, the coat is set by the fibre, not the bath. The coat's thickness in units of ℓ_c Ca^(2/3) against the fibre's radius in capillary lengths, on logarithmic axes: the dynamic meniscus's 1.3376 over the curvature the static meniscus asks of it, 1/b + Z/ℓ_c². Thin fibres take Quéré's 1.3376 b Ca^(2/3), set by their own radius; thick ones take Landau–Levich's 0.9458 ℓ_c Ca^(2/3). The two laws cross at b = 0.71 ℓ_c, where the coat is 0.61 of either; at b = 0.1 ℓ_c it is 0.137 of what the plate's law would give.

    A thin fibre coats by its own radius, and beads by it

    A plate drawn out of a bath carries a film set by the capillary length, the size at which surface tension and gravity balance. A fibre thinner than that length carries a film set by its own radius instead, often a tenth of what the plate's law promises — and the same radius then decides how quickly that film gathers into beads. The faster the fibre is drawn, the thicker its coat and the shorter the length of it that comes out smooth.

    part 2 · viscous

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